Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goalsmath121/Notes/notes10.pdfUnit #10...
Transcript of Unit #10 : Graphs of Antiderivatives, Substitution Integrals Goalsmath121/Notes/notes10.pdfUnit #10...
Unit #10 : Graphs of Antiderivatives, Substitution Integrals
Goals:
• Relationship between the graph of f (x) and its anti-derivative F (x)
• The guess-and-check method for anti-differentiation.
• The substitution method for anti-differentiation.
Graphs of Antiderivatives - 1
The Relation between the Integral and the Derivative Graphs
The Fundamental Theorem of Calculus states that∫ b
a
f (x) dx = F (b)− F (a)
if F (x) is an anti-derivative of f (x).Recognizing that finding anti-derivatives would be a central part of evaluatingintegrals, we introduced the notation∫
f (x) dx = F (x) ⇔ d
dxF (x) = f (x)
In cases where we might not be able to find an anti-derivative by hand, we can atleast sketch what the anti-derivative would look like; there are very clear relation-ships between the graph of f (x) and its anti-derivative F (x).
Graphs of Antiderivatives - 2
Example: Consider the graph of f (x) shown below. Sketch two possibleanti-derivatives of f (x).
0
2
4
6
−2
−4
1 2 3 4
x
f(x)
0
4
−4
1 2 3 4
x
F (x)
Graphs of Antiderivatives - 3
Example: Consider the graph of g(x) = sin(x) shown below. Sketch twopossible anti-derivatives of g(x).
x
g(x)
x
G(x)
Graphs of Antiderivatives - 4
The Fundamental Theorem of Calculus lets us add additional detail to the anti-derivative graph: ∫ b
a
f (x) dx = F (b)− F (a) = ∆F
What does this statement tell us about the graph of F (x) and f (x)?
Graphs of Antiderivatives - 5
Re-sketch the earlier anti-derivative graph of sin(x), find the area underneathone “arch” of the sine graph.
x
g(x)
x
G(x)
Antiderivative Graphs - Example 1 - 1
Example: Use the area interpretation of ∆F or F (b)− F (a) to constructa detailed sketch of the anti-derivative of f (x) for which F (0) = 2.
1
2
3
4
−1
−2
−3
−4
1 2 3 4−1−2−3−4
x
f(x)
0 1 2 3 4−1−2−3−4
x
F (x)
Antiderivative Graphs - Example 2 - 1
Example: f (x) is a continuous function, and f (0) = 1. The graph of f ′(x)is shown below.
0
1
2
−1
−2
1 2 3 4 5 6
x
f ′(x)
Based only on the information in the f ′ graph, identify the location and typesof all the critical points of f (x) on the interval x ∈ [0, 6].
Antiderivative Graphs - Example 2 - 2
Sketch the graph of f (x) on the interval [0, 6]. Reminder: f (0) = 1.
0
1
2
−1
−2
1 2 3 4 5 6
x
f ′(x)
0
1
2
−1
−2
1 2 3 4 5 6
x
f(x)
Antiderivative Graphs - Example 3 - 1
0 1 2 3 4 5
−2
−1
1
2
Question: Given the graph of f (x) above, which of the graphs below is ananti-derivative of f (x)?
0 1 2 3 4 5
−2
−1
1
2
3
4
5
6
0 1 2 3 4 5
−2
−1
1
2
3
4
5
6
0 1 2 3 4 5
−2
−1
1
2
3
4
5
6
0 1 2 3 4 5
−2
−1
1
2
3
4
5
6
(a) (b) (c) (d)
Integration Method - Guess and Check - 1
We now return to the challenge of finding a formula for an anti-derivative function.We saw simple cases last week, and now we will extend our methods to handle morecomplex integrals.
Anti-differentiation by Inspection: The Guess-and-Check Method
Often, even if we do not see an anti-derivative immediately, we can make an edu-cated guess and eventually arrive at the correct answer.
Integration Method - Guess and Check - 2
Example: Based on your knowledge of derivatives, what should the anti-
derivative of cos(3x),
∫cos(3x) dx, look most like?
(a) cos(x) + C
(b) sin(x) + C
(c) cos(3x) + C
(d) sin(3x) + C
Integration Method - Guess and Check - 3
Evaluate
∫cos(3x) dx by guessing and checking your antiderivative.
Integration Method - Guess and Check - 4
Example: Find
∫e7x−2 dx.
Integration Method - Guess and Check - 5
Example: Both of our previous examples had linear ‘inside’ functions. Hereis an integral with a quadratic ‘inside’ function:∫
xe−x2dx
Evaluate the integral.
Why was it important that there be a factor x in front of e−x2
in this integral?
Integration Method - Substitution - 1
Integration by Substitution
We can formalize the guess-and-check method by defining an intermediate variablethe represents the “inside” function.
Example: Show that
∫x3√x4 + 5 dx =
1
6(x4 + 5)3/2 + C.
Integration Method - Substitution - 2 ∫x3√x4 + 5 dx =
1
6(x4 + 5)3/2 + C
Relate this result to the chain rule.
Integration Method - Substitution - 3
Now use the method of substitution to evaluate
∫x3√x4 + 5 dx
Substitution Integrals - Example 1 - 1
Steps in the Method Of Substitution
1. Select a simple function w(x) that appears in the integral.
• Typically, you will also see w′ as a factor in the integrand as well.
2. Finddw
dxby differentiating. Write it in the form . . . dw = dx
3. Rewrite the integral using only w and dw (no x nor dx).
• If you can now evaluate the integral, the substitution was effective.
• If you cannot remove all the x’s, or the integral became harder instead of eas-ier, then either try a different substitution, or a different integration method.
Substitution Integrals - Example 1 - 2
Example: Find
∫tan(x) dx.
Substitution Integrals - Example 1 - 3
Though it is not required unless specifically requested, it can be reassuring to checkthe answer.Verify that the anti-derivative you found is correct.
Substitution Integrals - Example 2 - 1
Example: Find
∫x3ex
4−3dx.
Substitution Integrals - Example 3 - 1
Example: For the integral, ∫ex − e−x
(ex + e−x)2dx
both w = ex− e−x and w = ex + e−x are seemingly reasonable substitutions.Question: Which substitution will change the integral into the simpler form?
(a) w = ex − e−x
(b) w = ex + e−x
Substitution Integrals - Example 3 - 2
Compare both substitutions in practice.∫ex − e−x
(ex + e−x)2dx
with w = ex − e−x with w = ex + e−x
Substitution Integrals - Example 4 - 1
Example: What substitution is most likely to be helpful for evaluating theintegral ∫
sin(x)
1 + cos2(x)dx ?
(a) w = sin(x)
(b) w = cos(x)
(c) w = 1 + cos2(x)
Substitution Integrals - Example 4 - 2
Evaluate
∫sin(x)
1 + cos2(x)dx.
Substitutions and Definite Integrals - 1
Using the Method of Substitution for Definite Integrals
If we are asked to evaluate a definite integral such as∫ π/2
0
sinx
1 + cos2 xdx ,
where a substitution will ease the integration, we have two methods for handlingthe limits of integration (x = 0 and x = π/2).
a) When we make our substitution, convert both the variables x and the limits(in x) to the new variable; or
b) do the integration while keeping the limits explicitly in terms of x, writing thefinal integral back in terms of the original x variable as well, and then evaluating.
Substitutions and Definite Integrals - 2
Example: Use method a), converting the bounds to the new variable, toevaluate the integral ∫ π/2
0
sinx
1 + cos2 xdx
Substitutions and Definite Integrals - 3
Example: Use method b) method, keeping the bounds in terms of x, andconverting back to x’s, to evaluate∫ 64
9
√1 +√x√
xdx .
Simplifying Substitutions Integrals - 1
Simplifying Substitution Integrals
Sometimes trying a substitution will still simplify the integral, even if you don’tsee an obvious cue of “function and its derivative” in the integrand.Example: Find ∫
1√x + 1
dx .
Simplifying Substitutions Integrals - 2 ∫1√x + 1
dx